The complexity of finding maximum disjoint paths with length constraints

Itai, Alon; Perl, Yehoshua; Shiloach, Yossi

doi:10.1002/net.3230120306

Cited by 140 publications

(106 citation statements)

References 7 publications

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“…The ratio between the maximum number of node-disjoint s-t-paths and the size of a minimum length-bounded s-t-cut was also studied by Ben-Ameur [2000]. Itai et al [1982] give efficient algorithms to find the maximum number of nodeand edge-disjoint s-t-paths with at most 2 or 3 edges; the node-disjoint case is also solved for length-bound 4. On the complexity side they show that the maximum node-and edge-disjoint length-bounded s-t-paths problem is N P-hard for lengthbounds greater than 4.…”

Section: Introductionmentioning

confidence: 99%

Length-Bounded Cuts and Flows

Baier

Erlebach

Hall

et al. 2006

Automata, Languages and Programming

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For a given number L, an L-length-bounded edge-cut (node-cut, resp.) in a graph G with source s and sink t is a set C of edges (nodes, resp.) such that no s-t-path of length at most L remains in the graph after removing the edges (nodes, resp.) in C. An L-length-bounded flow is a flow that can be decomposed into flow paths of length at most L. In contrast to the classical flow theory, we describe instances for which the minimum L-length-bounded edge-cut (node-cut, resp.) is Θ(n 2/3 )-times (Θ( √ n)-times, resp.) larger than the maximum L-length-bounded flow, where n denotes the number of nodes; this is the worst case. We show that the minimum length-bounded cut problem is N P-hard to approximate within a factor of 1.1377 for L ≥ 5 in the case of nodecuts and for L ≥ 4 in the case of edge-cuts. We also describe algorithms with approximation ratio O(min{L, n/L}) ⊆ O( √ n) in the node case and O(min{L, n 2 /L 2 , √ m}) ⊆ O(n 2/3 ) in the edge case, where m denotes the number of edges. Concerning L-length-bounded flows, we show that in graphs with unit-capacities and general edge lengths it is N P-complete to decide whether there is a fractional length-bounded flow of a given value. We analyze the structure of optimal solutions and present further complexity results.

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Section: Introductionmentioning

confidence: 99%

Length-Bounded Cuts and Flows

Baier

Erlebach

Hall

et al. 2006

Automata, Languages and Programming

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“…We use a graph structure called lobe [31] to construct a graph from a 3SAT problem and on which finding two region-disjoint paths would provide a solution to that 3SAT problem. We will assume undirected networks, although directed links could also have been used.…”

Section: A Complexity Of the Problemmentioning

confidence: 99%

Finding Critical Regions and Region-Disjoint Paths in a Network

Trajanovski

Kuipers

Ilić

et al. 2015

IEEE/ACM Trans. Networking

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Abstract-Due to their importance to society, communication networks should be built and operated to withstand failures. However, cost considerations make network providers less inclined to take robustness measures against failures that are unlikely to manifest, like several failures coinciding simultaneously in different geographic regions of their network.Considering networks embedded in a two-dimensional plane, we study the problem of finding a critical region -a part of the network that can be enclosed by a given elementary figure of predetermined size -whose destruction would lead to the highest network disruption. We determine that only a polynomial, in the input, number of non-trivial positions for such a figure need to be considered and propose a corresponding polynomialtime algorithm. In addition, we consider region-aware network augmentation to decrease the impact of a regional failure. We subsequently address the region-disjoint paths problem, which asks for two paths with minimum total weight between a source (s) and a destination (d) that cannot both be cut by a single regional failure of diameter D (unless that failure includes s or d). We prove that deciding whether region-disjoint paths exist is NP-hard and propose a heuristic region-disjoint paths algorithm.

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“…If failures are expected to occur only sporadically (and in case of 1 : 1 protection), then it may be desirable to minimize the weight of the primary (shorter) path (min-min objective), which also leads to an NP-hard problem [109]. The min-max and minmin disjoint paths problems could be considered as extreme cases of the bounded disjoint paths problem, which was shown to be NP-hard [110] and later proven to be APX-hard by Bley [111] (the graph structure referred to as lobe that was used by Itai et al [110] to prove NP-completeness has since often been used to prove that other disjoint paths problems are NP-complete, e.g., [112][113][114]). Finding widest disjoint paths can easily be done by pruning "low-capacity" links from the graph and finding disjoint paths.…”

Section: Min-min Disjoint Paths Problemmentioning

confidence: 99%

An Overview of Algorithms for Network Survivability

Kuipers

2012

ISRN Communications and Networking

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Network survivability-the ability to maintain operation when one or a few network components fail-is indispensable for present-day networks. In this paper, we characterize three main components in establishing network survivability for an existing network, namely, (1) determining network connectivity, (2) augmenting the network, and (3) finding disjoint paths. We present a concise overview of network survivability algorithms, where we focus on presenting a few polynomial-time algorithms that could be implemented by practitioners and give references to more involved algorithms.

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The complexity of finding maximum disjoint paths with length constraints

Cited by 140 publications

References 7 publications

Length-Bounded Cuts and Flows

Length-Bounded Cuts and Flows

Finding Critical Regions and Region-Disjoint Paths in a Network

An Overview of Algorithms for Network Survivability

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